Factoring Trinomials

Multiply (x+3)(x+2)

Multiplying Binomials (FOIL)

F O I L

= x2 + 2x + 3x + 6

= x2 + 5x + 6

Distribute.2332 xxxx

x + 3

x

+

2

Using Algebra Tiles, we have:

= x2 + 5x + 6

Multiplying Binomials (Tiles)

Multiply (x+3)(x+2)

x2 x

x 1

x x

x

1 1

1 1 1

How can we factor trinomials such as x2 + 7x + 12 back into binomials?

One method is to again use algebra tiles:

1) Start with x2.

Factoring Trinomials (Tiles)

2) Add seven “x” tiles (vertical or horizontal, at

least one of each) and twelve “1” tiles.

x2 x x xxx

x

x

1 1 1

1 1 1

1 1

1 1

1 1

How can we factor trinomials such as x2 + 7x + 12 back into binomials?

One method is to again use algebra tiles:

1) Start with x2.

Factoring Trinomials (Tiles)

2) Add seven “x” tiles (vertical or horizontal, at

least one of each) and twelve “1” tiles.

x2 x x xxx

x

x

1 1 1

1 1 1

1 1

1 1

1 13) Rearrange the tiles until they form a rectangle!

We need to change the “x” tiles so the “1” tiles will fill in a rectangle.

How can we factor trinomials such as x2 + 7x + 12 back into binomials?

One method is to again use algebra tiles:

1) Start with x2.

Factoring Trinomials (Tiles)

2) Add seven “x” tiles (vertical or horizontal, at

least one of each) and twelve “1” tiles.

x2 x x xxx

x 1 1 1

1 1 1

1 1

1 1 1

1

3) Rearrange the tiles until they form a rectangle!

Still not a rectangle.

x

How can we factor trinomials such as x2 + 7x + 12 back into binomials?

One method is to again use algebra tiles:

1) Start with x2.

Factoring Trinomials (Tiles)

2) Add seven “x” tiles (vertical or horizontal, at

least one of each) and twelve “1” tiles.

x2 x x xx

x 1 1 1

1 1 1

1

1

1

1 113) Rearrange the tiles until they form a rectangle! A rectangle!!!

x

x

How can we factor trinomials such as x2 + 7x + 12 back into binomials?

One method is to again use algebra tiles:

4) Top factor:The # of x2 tiles = x’sThe # of “x” and “1” columns = constant.

Factoring Trinomials (Tiles)

5) Side factor:The # of x2 tiles = x’sThe # of “x” and “1” rows = constant.

x2 x x xx

x 1 1 1

1 1 1

1

1

1

1 11

x2 + 7x + 12 = ( x + 4)( x + 3)

x

x

x + 4

x

+

3

Again, we will factor trinomials such as x2 + 7x + 12 back into binomials.

This method does not use tiles, instead we look for the pattern of products and sums!

Factoring Trinomials (Method 2)

If the x2 term has no coefficient (other than 1)...

Step 1: List all pairs of numbers that multiply to equal the constant, 12.

x2 + 7x + 12

12 = 1 • 12

= 2 • 6

= 3 • 4

Factoring Trinomials (Method 2)

Step 2: Choose the pair that adds up to the middle coefficient.

x2 + 7x + 12

12 = 1 • 12

= 2 • 6

= 3 • 4

Step 3: Fill those numbers into the blanks in the binomials:

( x + )( x + )3 4

x2 + 7x + 12 = ( x + 3)( x + 4)

Factor. x2 + 2x - 24

This time, the constant is negative!

Factoring Trinomials (Method 2)

Step 1: List all pairs of numbers that multiply to equal the constant, -24. (To get -24, one number must be positive and one negative.)

-24 = 1 • -24, -1 • 24

= 2 • -12, -2 • 12

= 3 • -8, -3 • 8

= 4 • -6, - 4 • 6Step 2: Which pair adds up to 2?

Step 3: Write the binomial factors.

x2 + 2x - 24 = ( x - 4)( x + 6)

Factor. 3x2 + 14x + 8This time, the x2 term DOES have a coefficient (other than 1)!

Factoring Trinomials (Method 3*)

Step 2: List all pairs of numbers that multiply to equal that product, 24.

24 = 1 • 24

= 2 • 12

= 3 • 8

= 4 • 6Step 3: Which pair adds up to 14?

Step 1: Multiply 3 • 8 = 24 (the leading coefficient & constant).

a) Factor by Decomposition

Step 4: Rewrite the x term as two terms 82123 2 xxxStep 5: Divide the new polynomial into two pairs and factor out the common factor from each pair.

82123 2 xxx)4(2)4(3 xxx

)23)(4( xx

Factor. 2x2 - 7x -15This time, the x2 term DOES have a coefficient (other than 1)!

Factoring Trinomials (Method 2*)

Step 2: List all pairs of numbers that multiply to equal that product,

Step 3: Which pair adds up to ?

Step 1: Multiply (the leading coefficient & constant).

a) Factor by Decomposition

Step 4: Rewrite the x term as two terms

Step 5: Divide the new polynomial into two pairs and factor out the common factor from each pair.

Factor. 3x2 + 14x + 8This time, the x2 term DOES have a coefficient (other than 1)!

Factoring Trinomials (Method 2*)

Step 2: List all pairs of numbers that multiply to equal that product, 24.

24 = 1 • 24

= 2 • 12

= 3 • 8

= 4 • 6

Step 3: Which pair adds up to 14?

Step 1: Multiply 3 • 8 = 24 (the leading coefficient & constant).

b) Factor by Temporary Factors

( 3x + 2 )( x + 4 )

2

Factor. 3x2 + 14x + 8

Factoring Trinomials (Method 2*)

Step 5: Put the original leading coefficient (3) under both numbers.

( x + )( x + )

Step 6: Reduce the fractions, if possible.

Step 7: Move denominators in front of x.

Step 4: Write temporary factors with the two numbers.

123 3

2( x + )( x + )123 3

4

2( x + )( x + )43

( 3x + 2 )( x + 4 )

Factor. 3x2 + 14x + 8

Factoring Trinomials (Method 2*)

You should always check the factors by distributing, especially since this process has more than a couple of steps.

= 3x2 + 14 x + 8

= 3x • x + 3x • 4 + 2 • x + 2 • 4

√

3x2 + 14x + 8 = (3x + 2)(x + 4)

Factor 3x2 + 11x + 4

This time, the x2 term DOES have a coefficient (other than 1)!

Factoring Trinomials (Method 2*)

Step 2: List all pairs of numbers that multiply to equal that product, 12.

12 = 1 • 12

= 2 • 6

= 3 • 4

Step 3: Which pair adds up to 11?

Step 1: Multiply 3 • 4 = 12 (the leading coefficient & constant).

None of the pairs add up to 11, this trinomial can’t be factored; it is PRIME.

Factor 4x2 -16

This time, the Quadratic Equation is a Binomial with a Subtraction Sign!

Factoring A Difference of Squares (Method 3)

Step 2: List two brackets, one with an addition sign, one with a subtraction sign.

Step 3: Place the square roots in each bracket.

Step 1: Check to see that both terms are perfect squares.

Check your answer by expanding!

22 )2(4 xx 2)4(16

)( 2x 2x4 4

Factor each trinomial, if possible. The first four do NOT have leading coefficients, the last two DO have leading coefficients. Watch out for signs!!

1) x2 – 4x– 21

2) x2 + 12x + 32

3) x2 –10x + 24

4) x2 + 3x – 18

5) 2x2 + x – 21

6) 3x2 + 11x + 10

Factor These Trinomials!

Solution #1: x2 – 4x – 21

1) Factors of -21: 1 • -21, -1 • 213 • -7, -3 • 7

2) Which pair adds to (- 4)?

3) Write the factors.

x2 – 4x – 21 = (x + 3)(x - 7)

Solution #2: x2 + 12x + 32

1) Factors of 32: 1 • 322 • 164 • 8

2) Which pair adds to 12 ?

3) Write the factors.

x2 + 12x + 32 = (x + 4)(x + 8)

Solution #3: x2 - 10x + 24

1) Factors of 24: 1 • 242 • 123 • 84 • 6

2) Which pair adds to -10 ?

3) Write the factors.

x2 - 10x + 24 = (x - 4)(x - 6)

None of them adds to (-10). For the numbers to multiply to +24 and add to -10, they must both be negative!

-1 • -24-2 • -12-3 • -8-4 • -6

Solution #4: x2 + 3x - 18

1) Factors of -18: 1 • -18, -1 • 18 2 • -9, -2 • 93 • -6, -3 • 6

2) Which pair adds to 3 ?

3) Write the factors.

x2 + 3x - 18 = (x - 3)(x + 18)

Solution #5: 2x2 + x - 21

1) Multiply 2 • (-21) = - 42; list factors of - 42.

1 • -42, -1 • 42 2 • -21, -2 • 213 • -14, -3 • 146 • -7, -6 • 72) Which pair adds to 1 ?

3) Write the temporary factors.

2x2 + x - 21 = (x - 3)(2x + 7)

( x - 6)( x + 7)

4) Put “2” underneath.2 2

5) Reduce (if possible). ( x - 6)( x + 7)2 2

3

6) Move denominator(s)in front of “x”.

( x - 3)( 2x + 7)

Solution #6: 3x2 + 11x + 10

1) Multiply 3 • 10 = 30; list factors of 30.

1 • 302 • 153 • 105 • 62) Which pair adds to 11 ?

3) Write the temporary factors.

3x2 + 11x + 10 = (3x + 5)(x + 2)

( x + 5)( x + 6)

4) Put “3” underneath.3 3

5) Reduce (if possible). ( x + 5)( x + 6)3 3

2

6) Move denominator(s)in front of “x”.

( 3x + 5)( x + 2)